A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.


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1 The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty, you mght thnk ths pretty much wraps t up for classcal mechancs. And, n a sense, t does. Imagne a collecton of partcles, acted upon by a collecton of forces. The forces are added to gve a unque total force and Newtons famous F = ma s employed to fgure out where the partcles veloctes are takng t next. All you need s enough patence and a bg enough pece of paper and the problem s solved. However ths s hardly elegant and s to ad hoc to be the bass for the later development of quantum physcs and beyond. What s descrbed n these notes s based upon more powerful mathematcal technques and concepts developed by some of the gants, Euler, Lagrange, Hamlton and Jacob, of mathematcal physcs. Let us take the ld of Newton s three equatons of moton and look at what s nsde. The frst law: A partcle n a state of unform moton reman n that state of moton unless acted upon by external force. Newton recognses ths as the law of nerta of Galleo. I have used the word partcle nstead of body snce the dea of a pont partcle s more precse. It s magned, for example, that an elementary partcle such as an electron exsts at a pont n space and s not an extended object or body. In consderng the moton of the earth about the sun t s not a bad approxmaton to consder the earth as a pont partcle despte the evdent fact that t consst of an uncountable number of elementary partcles undergong some rather complcated motons whch are not at all descrbed by classcal mechancs. It s a part of our task to understand how s ths possble. From a modern pont of vew ths law has to do wth the defnton of an nertal frame S. Consder an elementary free test partcle wth some fxed mass. Here free means there s nothng else around whch mght effect ts moton. In an nertal farm t travels on the straght lne: r = r 0 + vt. (1) The exstence of such frames s a statement of Newton s frst law 1. An nertal frame s not unque. There are an nfnte number of nertal frames.let S be an nertal frame. Then there are 10 lnearly ndependent transformatons S S such that S s also an nertal frame (.e. f Eq. (1) holds n S, then t also holds n S ). To be precse: 1 Actually f you read, for example they tell you that you need three partcles wth such a moton n order to predct the same moton for a fourth partcle. 1
2 There are three statc rotatons: Three rotatons: r = Ô r (2) where Ô s a tme ndependent 3 3 orthogonal matrx, that s conventonally defned rotaton matrces. Three translatons: r = r + c (3) for a constant vector c. That s, there are translatons along x, y and z correspondng to the components of c and whch are ndependent. Three boosts: r = r + ut (4) for a constant velocty u. Ths just gves r 0 + vt r 0 + ( v + u)t. One tme Translaton: t = t + c (5) for a constant real number c. If moton s unform n S, t wll also be unform n S. These ten transformatons make up the Gallean Group under whch Newtons laws are nvarant. They wll be mportant later snce these symmetres of space and tme are the underlyng reason for conservaton laws. The rate of change of momentum ṗ s determned by the force F ( r, r) by: d p = F ( r, r) (6) In general, F ( r, r) can depend on both the poston r as well as the velocty r. Examples are frcton forces and the the v B Lorentz force. By defnton an object F whch rotates under rotatons Ô n the same manner as r s a vector. Even f the mass m m(t) s tme dependent and f p = m(t) v = m(t)d r/ then v s a vector and so s p and consequentally so s F Equaton (4) reduces to F = m a f ṁ = 0. But f m = m(t) (e.g. n rocket scence) then the Eq. (4) s expermentally the correct form. General theorems governng dfferental equatons guarantee that gven r and r at an ntal tme t = t 0, t s possble to ntegrate equaton Eq. (4) to determne r(t) for all t(as long as F s well behaved). Ths s the goal of classcal dynamcs. Newtons thrd law of moton: For every acton there s an equal and opposte reacton. 2
3 Ths nssts that all forces are due to nteractons between partcles and, n the end, reflects the conservaton of lnear momentum. Wth ths perspectve, for two partcles and j, t s requred that F j = F j (7) where F j s the force actng on partcle and due to partcle j. There s a second, stronger prncple, whch corresponds to the conservaton of angular momentum. The usual llustratons of someone leanng on a wall, that the wall pushes back wth the same force as that whch s appled reflects ths fact. In a more mathematcal fashon t s necessary to consder manypartcles. Now there s a label whch denotes whch partcle s beng consdered, that s: d p = F (8) where p s momentum of the partcle n queston and F the force actng upon t. It s assumed, supported ultmately by experment, that F = j F j + F ext that s, that there s a superposton prncple that the the force on a partcle s the sum of all ndvdual forces that would exst of any two partcles where studed n solaton. Here the system has been dvded nto the body and the sum j F j s over the forces between partcles n ths body whle F ext s the sum of forces due to all other bodes, Ths s an artfcal separaton. Now defne the total lnear momentum P gven by: then d P = P = d p = j,; j p (9) F j + Now a sum j,; j = 2 <j so, wth F ext F ext : Snce F j = F j, each ( F j + F j ) = 0 and F ext. dp = ( F j + F j ) + F ext (10) <j d P = F ext (11) Whch says that the rate of change of the total momentum of a body s determned only by the external forces whch act upon the body. Ths justfes usng Newton s second law for say a tenns ball. Defne the total mass M = m 3
4 where m s the mass of a gven partcle and the centre of mass: R = 1 m r M It follows f the m do not depend on tme so that P = d m d r = d m r = M d R F ext = M d2 R 2 (12) whch s our favourte F = m a. If the body s taken to be the whole unverse then d P = 0 whch s the mathematcal statement of the prncple of the conservaton of momentum. The total lnear momentum of the unverse s conserved. There s already a subtle mstake. It would usually be nssted that electrcal and magnetc forces gven by Maxwell s equatons can be ncluded n classcal mechancs. However t s also known that such felds carry momentum (and energy). The prncple of conservaton of lnear momentum defntely apples, but maybe no a smple statement of Newton s thrd law. The conon F j = F j s often called the weak verson of the thrd law. The strong verson nvolves the angular momentum defned by the usual l = r p (13) t follows usng p = m v that l = m r v (14) so that d l = md r d v d p v + m r = r snce d r = v by defnton and snce v v 0. Usng F = m a, ths can be rewrtten as d l = v F τ where τ s the torque. The result s then the famlar: τ = d l (15) 4
5 For many partcles, defned s L = l and so dl = r d p = r F j + F ext r F j + τ ext (16) j j j,; j where τ ext = τ ext = r F ext s the total external torque. The term j,; j r F j = <j ( r r j ) F j usng F j = F j. The strong verson of Newton s thrd law s satsfed when n adon to F j = F j t s also the case that ( r r j ) F j = 0 (17) <j For example, for the electrcal force F j = 1 q q j 4πɛ 0 r r j 3 ( r r j ) for whch F j = F j and for whch Eq.(17) s satsfed snce ( r r j ) ( r r j ) = 0. For such forces t follows d L = τ ext (18) and for the unverse as a whole d L = 0, correspondng to the statement The total angular momentum of the unverse s conserved. However there s a famous example of two partcles nteractng va the magnetc force F j = q v B j (19) where q s the charge of the partcle and where B j s the feld produced by another partcle. Consder the stuaton shown n the fgure Here partcle 2 produces no feld at partcle 1 and F 12 = 0 whereas partcle 1 produces a feld nto the paper at partcle 1 and a force F 21 = F 21 î. Here F j F j and Eq.(17) s not satsfed. Whle Newton s thrd law n ts smplest verson s lost. However the prncples of the conservaton of lnear and angular momentum are preserved snce the electromagnetc felds E and B n general carry both lnear and angular 5
6 momentum (the photon has spn). There s also energy assocated wth such felds. Consder next the energy of the system. The knetc energy s T = 1 2 mv2 = 1 2 md r d r and so dt = md r d v = d r d p = F d r usng the second law. The change T (t 2 ) T (t 1 ) = t2 t 1 dt = whch relates ths change n knetc energy to (20) F d r r2 = F d r (21) r 1 r2 r 1 F d r whch s the work done by the force. Notce what s dong the work and that the ntegral follows the path of the partcle. A conservatve force s one for whch r2 r 1 F d r s ndependent of the path taken between r 1 and r 2. Put n a postve sense F d r = 0 for any close path. Ths va Stoke s theorem s the equvalent to F = 0 everywhere. Defned s a potental functon: r0 V ( r) = F d r (22) r 6
7 where r 0 s an arbtrary pont at whch V ( r) = 0. It then follows that It s now the case that Eq. (21) reduces to or that that s there s an energy defned by F ( r) = V ( r). (23) T (t 2 ) T (t 1 ) = V ( r 2 ) V ( r 1 ) E = T (t 1 ) + V ( r 1 ) = T (t 2 ) + V ( r 2 ) (24) E = T + V (25) whch s a constant of moton. The Hamltonan, H( r, p), s defned by wrtng the energy n the form E( r, p) H. Thus n these rather lmted crcumstances The total energy of a partcle s conserved. There are nonconservatve forces, n partcular the Lorentz force q E for an electrc feld gven by Faraday s law: E = d B (26) corresponds to a F 0 f B s tme dependence. However the conservaton of total energy s a dearly held prncple. Agan the nonconservaton of mechancal energy n ths case has to do wth the fact that the electromagnetc feld carres energy and Faraday s law descrbes how ths energy s gven to, or taken from, the mechancal sector. Fnally snce the force can be wrtten as F ( r) = V ( r), Newton s second law reduces to d p = V ( r) (27) ths evdently assumng conservatve forces. 7
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